I have the exact same thing with mine.
It's a Dell Inspiron 6400, about three years old now. The power light always flashes (half the time red, half the time green) if it's plugged in, whether the computer's actually on or not.
Again, if I hover the mouse over the battery icon in the bottom corner it always says: "82% available (plugged in, charging)". However, if I unplug the computer from the socket, it automatically powers off, regardless of how much battery life it (suppossedly) has. It also won't work if it is not plugged in.
Several times when restarting it after such an instance when it powers off because I've unplugged it by accident, a message has popped up telling me to replace the battery as it's dead. However, as long as it's plugged in, it'll work. I don't understand this either.
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